
Standard Baseline: LR. This is the baseline model where no measures are taken against imbalance issues. For LR, we adopt a multiclass version of the logistic regression. For traditional datasets, LR is built upon a linear model. For deep learning datasets, LR is built upon a common deep backbone.

Over-sampling Methods. For these competitors, we first plug in an oversampling method to generate a more balanced dataset, then we use LR to train a model on the new dataset. Here we adopt the following two oversampling methods as our competitors:

BM. (BorderlineSMOTE) [1]}: This method is a variant of the SMOTE method (Synthetic Minority Over-sampling Technique), which restricts the oversampling process to the hard minority samples which lie at the borderline of the decision boundary.

MM. (MWMOTE) [2]}: MWMOTE is another variant of SMOTE. It first identifies the hard-to-learn informative minority class samples and assigns them weights according to their Euclidean distance from the nearest majority class samples.


Under-sampling Methods. For these competitors, we first adopt an undersampling method to generate a more balanced dataset, then we use LR to train a model on the new dataset. Here we adopt the following two methods as our competitors:

IHT. (InstanceHardnessThreshold) [3]}. This method works by filtering out the hard and noisy samples from the majority classes based on the instance hardness measure proposed in [3]}.

NM. (NearMiss)[5]}: It adopts an under-sampling idea to make majority class samples surround most of the minority class samples.

TL. (TomekLinks) [6]} It adopts the Tomek Links method to remove redundant samples that fail to contribute to the outline of the decision boundary from the majority class.


Imbalanced Loss Functions (for deep learning): For CIFAR-100-Imb and User-Imb dataset, we also compare our method with some of the recently proposed imbalanced loss functions for deep learning.

Focal Loss[7]}: It tackles the imbalance problem by adding a modulating factor to the cross-entropy loss to highlight the hard and minority samples during the training process.

CB-CE: It refers to the loss function that applies the reweighting scheme proposed in [8]} on the cross-entropy loss.

CB-Focal: It refers to the loss function that applies the reweighting scheme proposed in [8]} on the Focal loss.

LDAM[10]}: It proposes a Label-distribution-aware margin loss based on the minimum margin per class. Here we adopt the smooth relaxation for cross-entropy loss proposed therein.


Existing Deep AUC optimization methods:

DeepAUC: [11]} is a state-of-the-art stochastic AUC maximization algorithm developed for the deep neural network. It solves the AUC maximization from the saddle point problem. To ensure a fair comparison, we implement the algorithm that extends to the multi-class AUC problem by PyTorch. In addition, there are two optimization configurations in [11]}, including Proximal Primal-Dual Stochastic Gradient (PPD-SG) and Proximal Primal-Dual AdaGrad (PPD-AdaGrad). PPD-AdaGrad is employed in our experiment because it usually demonstrates better performance than PPD-SG in most cases.


Our Methods:

Ours1: An implementation of our learning framework with the square surrogate loss \(\ell _{sq}(\alpha ,t) = (\alpha - t)^2 \)

Ours2: An implementation of our learning framework with the exponential loss \( \ell _{\exp }(\alpha ,t) = \exp (-\alpha t)\)

Ours3: An implementation of our learning framework with the hinge loss \( \ell _{hinge}(\alpha ,t) = (\alpha - t)_+\)


